Can they do that? Would the speeding stuff even be admissible, as it was not included in the ticket, and is completely irrelevant pertaining to the case at hand (traffic control device Disobeyed 1110)?

Genuinely curious, as I'm no lawyer.

100% yes.

ever get a ticket, then when u are done watch a cop sit there for 20 minutes? hes not texting his wife, hes writing up every single last thing you do, what hes observing you to do, and everything about the situation.